Consider the two functions f(x,y) = 61/5 - 12x + x2 and g(x,y) = 15 - x2 - 6/5xy - 2/5y2 and let Q = {(x,y,z): f(x,y) <= z <= g(x,y)}.
a) Solve the equation f(x,y) = g(x,y) for y in terms of x, find the range of values of x for which the two solutions functions are defined, and plot these functions.
b) Use Mathematica to find the volume of Q by means of a double integral and time this calculation.
c) The orthogonal projection R of Q in the xy-plane is, of course, an elliptical region. Find a change of variables so that axes of this ellipse are parallel to the new coordinate axes. Find the Jacobian of this transformation, use the change of variables formula to re-calculate the volume of Q, and compare the calculation time for this method with that of part b.
Any help with this would be soooo much appreciated! I'm simply stuck and can't find any help anywhere.